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Deterministic Public Key Encryption and Identity-Based Encryption from Lattices in the Auxiliary-Input Setting.

Xiang Xie, Rui Xue, Rui Zhang

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Abstract. Deterministic public key encryption (D-PKE) provides an alternative to randomized public key encryption in various scenarios (e.g. search on encrypted data) where the latter exhibits inherent drawbacks. In CRYPTO’11, Brakerski and Segev formalized a framework for studying the security of deterministic public key encryption schemes with respect to auxiliary inputs. A trivial requirement is that the plaintext should not be efficiently recoverable from the auxiliary inputs. In this paper, we present an efficient deterministic public key encryption scheme in the auxiliary-input setting from lattices. The public key size, ciphertext size and ciphertext expansion factor are improved compared with the scheme proposed by Brakerski and Segev. Our scheme is also secure even in the multi-user setting where related messages may be encrypted under multiple public keys. In addition, the security of our scheme is based on the hardness of the learning with errors (LWE) problem which remains hard even for quantum algorithms. Furthermore, we consider deterministic identity-based public key encryption (D-IBE) in the auxiliaryinput setting. The only known D-IBE scheme (without considering auxiliary inputs) in the standard model was proposed by Bellare et al. in EUROCRYPT’12. However, this scheme is only secure in the selective security setting, and Bellare et al. identified it as an open problem to construct adaptively secure D-IBE schemes. The second contribution of this work is to propose a D-IBE scheme from lattices that is adaptively secure.

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Applied Mathematics & Information Sciences•Oct 9, 2013•0 citations
Abstract. Deterministic public key encryption (D-PKE) provides an alternative to randomized public key encryption in various scenarios (e.g. search on encrypted data) where the latter exhibits inherent drawbacks. In CRYPTO’11, Brakerski and Segev formalized a framework for studying the security of deterministic public key encryption schemes with respect to auxiliary inputs. A trivial requirement is that the plaintext should not be efficiently recoverable from the auxiliary inputs. In this paper, we present an efficient deterministic public key encryption scheme in the auxiliary-input setting from lattices. The public key size, ciphertext size and ciphertext expansion factor are improved compared with the scheme proposed by Brakerski and Segev. Our scheme is also secure even in the multi-user setting where related messages may be encrypted under multiple public keys. In addition, the security of our scheme is based on the hardness of the learning with errors (LWE) problem which remains hard even for quantum algorithms. Furthermore, we consider deterministic identity-based public key encryption (D-IBE) in the auxiliaryinput setting. The only known D-IBE scheme (without considering auxiliary inputs) in the standard model was proposed by Bellare et al. in EUROCRYPT’12. However, this scheme is only secure in the selective security setting, and Bellare et al. identified it as an open problem to construct adaptively secure D-IBE schemes. The second contribution of this work is to propose a D-IBE scheme from lattices that is adaptively secure.
Proxy Re-Encryption (PRE) is a cryptographic primitive that allows a proxy to turn an Alice’s ciphertext into a Bob’s ciphertext on the same plaintext. All of the PRE schemes are public key encryption and semantic security. Deterministic Public Key Encryption (D-PKE) provides an alternative to randomized public key encryption in various scenarios where the latter exhibits inherent drawbacks. In this paper, we construct the first multi-use unidirectional D-PRE scheme from Lattices in the auxiliary-input setting. We also prove that it is PRIV1-INDr secure in the standard model based on the LWR. Finally, an identity-based D-PRE is obtained from the basic construction.
Using a known lattice-based chosen plain-text secure encryption scheme and the Bonsai trees primitive, a identity-based encryption (IBE) scheme is proposed in this paper. A public matrices chosen rule that has been used in a known efficient lattice-based chosen ciphtertext secure encryption scheme is used in this paper to reduce the public key size and the message-to ciphtertext expanse factor. Under the hardness of the decision variant of the learning with errors (LWE) problem, we prove that the proposed IBE scheme is indistinguishable against the adaptive chosen identities and chosen-plaintext attack in the standard model. Since a new public key chosen rule is used in our design scheme, the public key size of the proposed scheme would be short. Moreover, the message-to-ciphtertext expanse factor of this scheme is also controlled efficiently which is nearly closed to that of Gentry's scheme. Due to the quantum intractability of the LWE problem on which the scheme is based, the proposed IBE scheme is secure even in quantum-era.
An efficient identity-based encryption (IBE) scheme over lattice is proposed in this paper. Under the hardness of the learning with errors (LWE) problem, the proposed scheme is semantic secure against adaptive chosen identity and chosen plaintext attack in the standard model. To improve the efficiency of the lattice-based IBE scheme, unlike the identity string is encoded into a matrix by a group of public matrices in several known constructions, the identity string of l bits is encoded into a vector with the help of l+1 vectors in this paper. With the help of this idea, we achieve the private key extraction of IBE scheme at the same lattice. Then, the public key of the proposed scheme only consists of one n×m matrix and l+1 vectors, compared with that the public keys of the known lattice-based IBE schemes all consist as a group of n×m matrices. Hence, the public key size of this scheme is shorter than that of the known constructions.
Using lattice basis delegation in a fixed dimension, we propose an efficient lattice-based hierarchical identity based encryption (HIBE) scheme in the standard model whose public key size is only ( dm 2 + mn ) log q bits and whose message-ciphertext expansion factor is only log q , where d is the maximum hierarchical depth and ( n , m , q ) are public parameters. In our construction, a novel public key assignment rule is used to averagely assign one random and public matrix to two identity bits, which implies that d random public matrices are enough to build the proposed HIBE scheme in the standard model, compared with the case in which 2 d such public matrices are needed in the scheme proposed at Crypto 2010 whose public key size is (2 dm 2 + mn + m ) log q . To reduce the message-ciphertext expansion factor of the proposed scheme to log q , the encryption algorithm of this scheme is built based on Gentry’s encryption scheme, by which m 2 bits of plaintext are encrypted into m 2 log q bits of ciphertext by a one time encryption operation. Hence, the presented scheme has some advantages with respect to not only the public key size but also the message-ciphertext expansion factor. Based on the hardness of the learning with errors problem, we demonstrate that the scheme is secure under selective identity and chosen plaintext attacks.
There only exists one deterministic identity-based encryption (DIBE) scheme which is adaptively secure in the auxiliary-input setting, under the learning with errors (LWE) assumption. However, the master public key consists of basic matrices. In this paper, we consider to construct adaptively secure DIBE schemes with more compact public parameters from the LWE problem. (i) On the one hand, we gave a generic DIBE construction from lattice-based programmable hash functions with high min-entropy. (ii) On the other hand, when instantiating our generic DIBE construction with four LPHFs with high min-entropy, we can get four adaptively secure DIBE schemes with more compact public parameters. In one of our DIBE schemes, the master public key only consists of basic matrices.
Abstract: Using the Bonsai trees primitive and Gentry’s CPA-secure (chosen-plaintext attack) public-key encryption (PKE) scheme, we propose an efficient chosen-ciphtertext secure PKE scheme over lattice. If the decision variant of the learning with errors (LWE) problem is hard and the one-time signature used in this scheme is strong unforgeable, the proposed PKE scheme is indistinguishable against the adaptive chosen-ciphtertext attack (IND-CCA2). One of the characters for this scheme is that, before any encryption operation, the encryption algorithm uses a new choice rule to fix the public parameter matrixes used in the encryption operation. With the help of this new choice rule, we can achieve the chosen-ciphtertext security with much shorter the public key size in contrast to the lattice-based encryption scheme proposed in STOC’09 by Peikert. Moreover, as a CCA-secure PKE scheme, the message-tociphtertext expanse factor of this scheme which is controlled efficiently is nearly closed to the message-to-ciphtertext expanse factor of Gentry’s scheme which is CPA secure. Due to the quantum intractability of the LWE problem on which the scheme is based, the proposed PKE scheme is secure even in quantum-era.